Diophantine sets are subsets defined by polynomial equations over integers. More precisely We obtain them basically by projecting integers solutions of some polynomial equations. Observation 1: Using the statement iff we see that there is no difference between one polynomial equation and a system of polynomial equations. Example: Even numbers are Diophantine: Odd numbers: Non-negative […]
Category: Number Theory
Mordell’s Equation
The equation is called Mordell’s equation. For , it is non-singular and defines an elliptic curve (genus 1, group structure). Rational Points: We first describe the torsion part of the rational solutions. By Nagell-Lutz, these solutions should have integer coordinates. Torsion: Assume that is free of sixth powers. If not we can use the change […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Kürschâk and Nagel’s theorems (Erdos 1932)
Consider the familiar reciprocal sums None of the above quantities are integers.The first, second, and fourth cases all follow from one very elementary principle. One looks for a prime which occurs in one denominator more strongly than it occurs in every other denominator. After the fractions are put over a common denominator, every term except […]
Betrand Postulate : Erdos( 1932)
Bertrand’s postulate states that for every integer , there is a prime satisfying The statement is elementary, but it is remarkably strong: no matter how far one goes along the number line, one never encounters a multiplicative gap as large as a factor of containing no primes. Erdős’s proof (1932) of this fact is centered […]
Zeta(2)
Basel Problem (1644) asks to find the exact value of the series Euler (1735) showed that At first glance, this is a problem about a list of numbers. Yet its answer, contains , a constant associated with circles, periodicity, and geometry. The surprise is not merely that the sum has a closed form. It is […]
Eisenstein’s Lattice Point Proof of Quadratic Reciprocity
The Law of Quadratic Reciprocity is one of the central results of classical number theory. Gauss famously called it the “Theorema Aureum,” or Golden Theorem. It reveals a hidden symmetry between two different modular worlds. At first glance, the question “Is a square modulo ?” seems unrelated to the question “Is a square modulo ?” […]
A Neighbourhood Of Infinity 